#367113

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(2,3)(5,6)(7,8)(9,10,11,12)(13,14)(15,23)(16,20)(17,26)(18,22)(19,21)(24,27)(25,28);(1,2,4,3)(5,6)(7,8)(13,15,17,25,18,16,19,24)(14,20,26,27,22,23,21,28);(6,8)(10,12)(13,16,18,15)(14,20,22,23)(17,25,19,24)(21,27,26,28);(1,3,4,2)(13,17,18,19)(14,21,22,26)(15,24,16,25)(20,27,23,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274309);